# nLab differential algebroid

Contents

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Algebra

higher algebra

universal algebra

categorification

# Contents

## Idea

A differential algebroid is an oidification of the concept of differential algebra.

## Definitions

Let $K$ be a commutative ring and $C$ be a $K$-linear category. Then $C$ is a $K$-differential algebroid if for every hom-$K$-module $Mor(a,b)$ there is a $K$-linear morphism $d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b)$ such that for all objects $a,b,c\in Ob(C)$, morphisms $f:Mor(a,b)$ and $g:Mor(b,c)$, and $K$-linear morphisms $d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b)$, $d_{Mor(b,c)}:Mor(b,c) \to Mor(b,c)$, and $d_{Mor(a,c)}:Mor(a,c) \to Mor(a,c)$, a generalised Leibniz rule is satisfied:

$d_{Mor(a,c)}(g \circ f) = d_{Mor(b,c)}(g) \circ f + g \circ d_{Mor(a,b)}(f) \,.$

If all three objects are the same, this reduces down to the Leibniz rule for a derivation.

## Examples

Last revised on May 23, 2021 at 21:10:36. See the history of this page for a list of all contributions to it.